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.. _math:
==============
Math Functions
==============
.. include:: ../check.rst
.. raw:: html
<style> .green {color:green} </style>
.. role:: green
.. toctree::
:hidden:
log.rst
stdfix.rst
.. contents:: Table of Contents
:depth: 4
:local:
Source Locations
================
- The main source is located at: `libc/src/math <https://github.com/llvm/llvm-project/tree/main/libc/src/math>`_.
- The tests are located at: `libc/test/src/math <https://github.com/llvm/llvm-project/tree/main/libc/test/src/math>`_.
- The floating point utilities are located at: `libc/src/__support/FPUtil <https://github.com/llvm/llvm-project/tree/main/libc/src/__support/FPUtil>`_.
Implementation Requirements / Goals
===================================
* The highest priority is to be as accurate as possible, according to the C and
IEEE 754 standards. By default, we will aim to be correctly rounded for `all rounding modes <https://en.cppreference.com/w/c/numeric/fenv/FE_round>`_.
The current rounding mode of the floating point environment is used to perform
computations and produce the final results.
- To test for correctness, we compare the outputs with other correctly rounded
multiple-precision math libraries such as the `GNU MPFR library <https://www.mpfr.org/>`_
or the `CORE-MATH library <https://core-math.gitlabpages.inria.fr/>`_.
* Our next requirement is that the outputs are consistent across all platforms.
Notice that the consistency requirement will be satisfied automatically if the
implementation is correctly rounded.
* Our last requirement for the implementations is to have good and predicable
performance:
- The average performance should be comparable to other ``libc``
implementations.
- The worst case performance should be within 10X-20X of the average.
- Platform-specific implementations or instructions could be added whenever it
makes sense and provides significant performance boost.
* For other use cases that have strict requirements on the code size, memory
footprint, or latency, such as embedded systems, we will aim to be as accurate
as possible within the memory or latency budgets, and consistent across all
platforms.
Add a new math function to LLVM libc
====================================
* To add a new math function, follow the steps at: `libc/src/math/docs/add_math_function.md <https://github.com/llvm/llvm-project/tree/main/libc/src/math/docs/add_math_function.md>`_.
Implementation Status
=====================
* To check math functions enabled for Linux:
- `linux-x86_64 <https://github.com/llvm/llvm-project/tree/main/libc/config/linux/x86_64/entrypoints.txt>`_
- `linux-aarch64 <https://github.com/llvm/llvm-project/tree/main/libc/config/linux/aarch64/entrypoints.txt>`_
- `linux-aarch32 <https://github.com/llvm/llvm-project/tree/main/libc/config/linux/arm/entrypoints.txt>`_
- `linux-riscv64 <https://github.com/llvm/llvm-project/tree/main/libc/config/linux/riscv64/entrypoints.txt>`_
* To check math functions enabled for Windows:
- `windows-x86_64 <https://github.com/llvm/llvm-project/tree/main/libc/config/windows/entrypoints.txt>`_
- windows-aarch64 - to be added
* To check math functions enabled for macOS:
- `darwin-x86_64 <https://github.com/llvm/llvm-project/tree/main/libc/config/darwin/x86_64/entrypoints.txt>`_
- `darwin-aarch64 <https://github.com/llvm/llvm-project/tree/main/libc/config/darwin/arm/entrypoints.txt>`_
* To check math functions enabled for GPU:
- `gpu-entrypoints <https://github.com/llvm/llvm-project/tree/main/libc/config/gpu/entrypoints.txt>`_
* To check math functions enabled for embedded system:
- `baremetal-aarch32 <https://github.com/llvm/llvm-project/tree/main/libc/config/baremetal/arm/entrypoints.txt>`_
- baremetal-riscv32 - to be added
Basic Operations
----------------
+------------------+---------------------------------------+-------------------+-------------------+-------------------+-------------------+
| <Func> | Linux | Windows | MacOS | Embedded | GPU |
| +---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| | x86_64 | aarch64 | aarch32 | riscv64 | x86_64 | aarch64 | x86_64 | aarch64 | aarch32 | riscv32 | AMD | nVidia |
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| nexttowardf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nexttowardl | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nextup | |check| | |check| | |check| | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nextupf | |check| | |check| | |check| | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nextupl | |check| | |check| | |check| | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nextupf128 | |check| | |check| | | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| remainder | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| remainderf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| remainderl | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| remquo | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| remquof | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| remquol | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| rint | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| rintf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| rintl | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| rintf128 | |check| | |check| | | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| round | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| roundf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| roundl | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| roundf128 | |check| | |check| | | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| scalbn | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| scalbnf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| scalbnl | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| trunc | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| truncf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| truncl | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| truncf128 | |check| | |check| | | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ufromfp | |check| | |check| | |check| | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ufromfpf | |check| | |check| | |check| | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ufromfpl | |check| | |check| | |check| | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ufromfpf128 | |check| | |check| | | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ufromfpx | |check| | |check| | |check| | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ufromfpxf | |check| | |check| | |check| | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ufromfpxl | |check| | |check| | |check| | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ufromfpxf128 | |check| | |check| | | |check| | | | | | | | | |
+------------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
Higher Math Functions
---------------------
+------------+---------------------------------------+-------------------+-------------------+-------------------+-------------------+
| <Func> | Linux | Windows | MacOS | Embedded | GPU |
| +---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| | x86_64 | aarch64 | aarch32 | riscv64 | x86_64 | aarch64 | x86_64 | aarch64 | aarch32 | riscv32 | AMD | nVidia |
+============+=========+=========+=========+=========+=========+=========+=========+=========+=========+=========+=========+=========+
| acos | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| acosf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| acosl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| acosh | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| acoshf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| acoshl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| asin | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| asinf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| asinl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| asinh | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| asinhf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| asinhl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atan | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atanf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atanl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atan2 | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atan2f | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atan2l | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atanh | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atanhf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atanhl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| cbrt | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| cbrtf | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| cbrtl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| cos | |check| | | | | |check| | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| cosf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| cosl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| cosh | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| coshf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| coshl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erf | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erff | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erfl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erfc | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erfcf | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erfcl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| exp | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| expf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| expl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| exp10 | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| exp10f | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| exp10l | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| exp2 | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| exp2f | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| exp2l | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| expm1 | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| expm1f | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| expm1l | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fma | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fmaf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fmal | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| hypot | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| hypotf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| hypotl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| lgamma | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| lgammaf | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| lgammal | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| logf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| logl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log10 | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log10f | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log10l | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log1p | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log1pf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log1pl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log2 | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log2f | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log2l | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| pow | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| powf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| powl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sin | |check| | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sinf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sinl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sincos | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sincosf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sincosl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sinh | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sinhf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sinhl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sqrt | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sqrtf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sqrtl | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sqrtf128 | |check| | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tan | |check| | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tanf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tanl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tanh | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tanhf | |check| | |check| | |check| | |check| | |check| | | | |check| | |check| | |check| | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tanhl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tgamma | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tgammaf | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tgammal | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
Accuracy of Higher Math Functions
=================================
============== ================ =============== ====================== ======================
<Func> <Func_f> (float) <Func> (double) <Func_l> (long double) <Func_f128> (float128)
============== ================ =============== ====================== ======================
acos |check|
acosh |check|
asin |check|
asinh |check|
atan |check|
atanh |check|
cos |check| large
cosh |check|
erf |check|
exp |check| |check|
exp10 |check| |check|
exp2 |check| |check|
expm1 |check| |check|
fma |check| |check|
hypot |check| |check|
log |check| |check|
log10 |check| |check|
log1p |check| |check|
log2 |check| |check|
pow |check|
sin |check| large
sincos |check| large
sinh |check|
sqrt |check| |check| |check| |check|
tan |check|
tanh |check|
============== ================ =============== ====================== ======================
Legends:
* |check| : correctly rounded for all 4 rounding modes.
* CR: correctly rounded for the default rounding mode (round-to-the-nearest,
tie-to-even).
* x ULPs: largest errors recorded.
..
TODO(lntue): Add a new page to discuss about the algorithms used in the
implementations and include the link here.
Performance
===========
* Simple performance testings are located at: `libc/test/src/math/performance_testing <https://github.com/llvm/llvm-project/tree/main/libc/test/src/math/performance_testing>`_.
* We also use the *perf* tool from the `CORE-MATH <https://core-math.gitlabpages.inria.fr/>`_
project: `link <https://gitlab.inria.fr/core-math/core-math/-/tree/master>`_.
The performance results from the CORE-MATH's perf tool are reported in the
table below, using the system library as reference (such as the `GNU C library <https://www.gnu.org/software/libc/>`_
on Linux). Fmod performance results obtained with "performance_testing".
+--------------+-------------------------------+-------------------------------+-------------------------------------+----------------------------------------------------------------------+
| <Func> | Reciprocal throughput (clk) | Latency (clk) | Testing ranges | Testing configuration |
| +-----------+-------------------+-----------+-------------------+ +-------------+-------------------------+--------------+---------------+
| | LLVM libc | Reference (glibc) | LLVM libc | Reference (glibc) | | CPU | OS | Compiler | Special flags |
+==============+===========+===================+===========+===================+=====================================+=============+=========================+==============+===============+
| acosf | 24 | 29 | 62 | 77 | :math:`[-1, 1]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| acoshf | 18 | 26 | 73 | 74 | :math:`[1, 21]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| asinf | 23 | 27 | 62 | 62 | :math:`[-1, 1]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| asinhf | 21 | 39 | 77 | 91 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| atanf | 27 | 29 | 79 | 68 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| atanhf | 18 | 66 | 68 | 133 | :math:`[-1, 1]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| cosf | 13 | 32 | 53 | 59 | :math:`[0, 2\pi]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| coshf | 14 | 20 | 50 | 48 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| expf | 9 | 7 | 44 | 38 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| exp10f | 10 | 8 | 40 | 38 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| exp2f | 9 | 6 | 35 | 31 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| expm1f | 9 | 44 | 42 | 121 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| fmodf | 73 | 263 | - | - | [MIN_NORMAL, MAX_NORMAL] | i5 mobile | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
| +-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| | 9 | 11 | - | - | [0, MAX_SUBNORMAL] | i5 mobile | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| fmod | 595 | 3297 | - | - | [MIN_NORMAL, MAX_NORMAL] | i5 mobile | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
| +-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| | 14 | 13 | - | - | [0, MAX_SUBNORMAL] | i5 mobile | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| hypotf | 25 | 15 | 64 | 49 | :math:`[-10, 10] \times [-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| logf | 12 | 10 | 56 | 46 | :math:`[e^{-1}, e]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| log10f | 9 | 17 | 35 | 48 | :math:`[e^{-1}, e]` | Ryzen 5900X | Ubuntu 22.04 LTS x86_64 | Clang 15.0.6 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| log1pf | 16 | 33 | 61 | 97 | :math:`[e^{-0.5} - 1, e^{0.5} - 1]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| log2f | 13 | 10 | 57 | 46 | :math:`[e^{-1}, e]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| sinf | 12 | 25 | 51 | 57 | :math:`[-\pi, \pi]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| sincosf | 19 | 30 | 57 | 68 | :math:`[-\pi, \pi]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| sinhf | 13 | 63 | 48 | 137 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| tanf | 16 | 50 | 61 | 107 | :math:`[-\pi, \pi]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| tanhf | 13 | 55 | 57 | 123 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
Algorithms + Implementation Details
===================================
* :doc:`log`
Fixed-point Arithmetics
=======================
* :doc:`stdfix`
References
==========
* `CRLIBM <https://hal-ens-lyon.archives-ouvertes.fr/ensl-01529804/file/crlibm.pdf>`_.
* `RLIBM <https://people.cs.rutgers.edu/~sn349/rlibm/>`_.
* `Sollya <https://www.sollya.org/>`_.
* `The CORE-MATH Project <https://core-math.gitlabpages.inria.fr/>`_.
* `The GNU C Library (glibc) <https://www.gnu.org/software/libc/>`_.
* `The GNU MPFR Library <https://www.mpfr.org/>`_.
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